Rarely. I doubt there is a better condition that guarantees this than "that the two sums are equal".
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Alex BeckerSep 14 '12 at 17:23

Why is it that you want to know - is there some context to the question?
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Mark BennetSep 14 '12 at 17:25

2

It would be a more natural question, to my way of thinking, if the left-hand-side (which is the sum of $n$ fractions) were divided by $n$ to match the single fraction on the right-hand-side.
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Mark BennetSep 14 '12 at 17:28

The sum on the right side of the equality is equal to each of the fractions on the left if all of those are equal to each other. But that's a different question.
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Michael HardySep 14 '12 at 17:29

2

I would recommend that before you worry about infinite summations you solve the case $n=2$. What do you get if you start with $(a/b)+(c/d)=(a+c)/(b+d)$?
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Gerry MyersonSep 17 '12 at 12:53

The equality of $(*)$ holds when $a_1/b_1=\cdots=a_n/b_n$, and
the equality of $(**)$ holds when at most one of $a_i$s are nonzero;
this suggests the equality of $(*)$ and $(**)$ hold at the same time only when all $a_i$s are zero.